Comment by foxes
3 days ago
I feel its not that bad an abuse of notation as kinda consistent with other areas of mathematics -
A coset, quotients r + I, affine subspaces v + W, etc. Not literal sets but comparing some representative with a class label, and the `=, +` is defined not just on the actual objects but on some other structure used to make some comparison too.
No. Equality is defined on sets, with exactly one exception, this stupid fucking notation.
Notation is invented to be useful. A lot of mathematicians and physicists write stuff like
f = g + O(h)
and then proceed to do a bunch of manipulations without getting bogged down in set theory syntax.
Also equality isnt just defined on sets, usually people start by defining it as like a formal thing satisfying a bunch of rules as a part of their logic, and going from there.